As \(\alpha _{_{1,5}}=\alpha _{_{2,4}}=0\) , the PDEs above are changed into
the first type Appell PDEs. The holonomic rank of the ideal composed by the
partial differential operators of above PDEs is 3, which indicates the fundamental
solution system is constituted by three linear independent hypergeometric functions
in certain neighborhoods of regular singular points.
It is known that the fourth type Appell function cannot be imbedded into
the Grassmannians [1]}. Nevertheless, including the appropriate power factor
and the corresponding transformation of the variables, the Feynman integral presented through
the fourth Appell functions [2]}, [3]}, [4]} satisfies the above PDEs.
Those matrices of integer lattice permit 12 possibilities of the exponents.
